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A273898 Sum of the abscissae of the first descents of all bargraphs of semiperimeter n (n>=2).

%I A273898 #26 Jul 24 2022 12:30:16
%S A273898 1,3,9,27,81,244,739,2251,6895,21232,65703,204245,637573,1997892,
%T A273898 6282635,19820580,62716923,198997349,633015543,2018391204,6449819095,
%U A273898 20652628601,66256638509,212939343591,685497649231,2210217592624,7136781993563,23076554161563
%N A273898 Sum of the abscissae of the first descents of all bargraphs of semiperimeter n (n>=2).
%C A273898 A descent in a bargraph is a maximal sequence of adjacent down steps.
%H A273898 Alois P. Heinz, <a href="/A273898/b273898.txt">Table of n, a(n) for n = 2..1000</a>
%H A273898 M. Bousquet-Mélou and A. Rechnitzer, <a href="http://dx.doi.org/10.1016/S0196-8858(02)00553-5">The site-perimeter of bargraphs</a>, Adv. in Appl. Math. 31 (2003), 86-112.
%H A273898 Emeric Deutsch, S Elizalde, <a href="http://arxiv.org/abs/1609.00088">Statistics on bargraphs viewed as cornerless Motzkin paths</a>, arXiv preprint arXiv:1609.00088, 2016
%F A273898 G.f.:  g(z)=(1-4z+3z^2-(1-2z)Q)/(2z^3), where Q = sqrt(1-4z+2z^2+z^4).
%F A273898 a(n) = Sum(k*A273897(n,k), k>=1).
%F A273898 a(n) = A082582(n+2)-2*A082582(n+1).
%F A273898 D-finite with recurrence (n+3)*a(n) +2*(-3*n-4)*a(n-1) +2*(5*n-2)*a(n-2) +4*(-n+2)*a(n-3) +(n-3)*a(n-4) +2*(-n+5)*a(n-5)=0. - _R. J. Mathar_, Jul 24 2022
%e A273898 a(4)=9 because the 5 (=A082582(4)) bargraphs of semiperimeter 4 correspond to the compositions [1,1,1],[1,2],[2,1],[2,2],[3] and the corresponding pictures give the values 3,2,1,2,1 for the abscissae of their first descents.
%p A273898 g := ((1-4*z+3*z^2-(1-2*z)*Q)*(1/2))/z^3: Q := sqrt(1-4*z+2*z^2+z^4): gser := series(g,z = 0,40): seq(coeff(gser, z, n), n = 2 .. 35);
%p A273898 # second Maple program:
%p A273898 a:= proc(n) option remember; `if`(n<4, [0$2, 1, 3][n+1],
%p A273898      ((2*(14*n^2+6+13*n))*a(n-1)-(2*(7*n^2-6-4*n))*a(n-2)
%p A273898      +12*a(n-3) -(n-4)*(3+7*n)*a(n-4))/((n+3)*(7*n-4)))
%p A273898     end:
%p A273898 seq(a(n), n=2..40);  # _Alois P. Heinz_, Jun 07 2016
%t A273898 a[n_] := a[n] = If[n<4, {0, 0, 1, 3}[[n+1]], ((2*(14*n^2+6+13*n))*a[n-1] - (2*(7*n^2-6-4*n))*a[n-2] + 12*a[n-3] - (n-4)*(3+7*n)*a[n-4])/((n+3)*(7*n - 4))]; Table[a[n], {n, 2, 40}] (* _Jean-François Alcover_, Dec 02 2016 after _Alois P. Heinz_ *)
%Y A273898 Cf. A082582, A273897.
%K A273898 nonn
%O A273898 2,2
%A A273898 _Emeric Deutsch_, Jun 06 2016